String stability for vehicular
platoon control
Robson and Shahriar
Motivation
The aim of this project will be to discuss the analysis of string stability of a set of
networked cars over CACC (Cooperative Adaptive Cruise Control). Today, there
are many car accidents happening in the world. 1.35 million people are killed on
roadways around the world every year with almost 3,700 deaths daily
1
. This
makes car accident the leading cause of death for children and young people
5–29 years of age
1
. It is estimated car crashes will cost the world economy
approximately $1.8 trillion 2015-2030. With the advances in technology and
engineering, we can use now autonomous cars including driver assistance
systems and connected vehicles to reduce accidents. In addition, vehicle
platooning has received significant research attention due to its potential in
reducing traffic congestion. However, there remains the big question of stability in
the design and development of these systems. Guaranteeing the stability of these
systems by in large ensures the safety of these systems. As such, this projects
aims to address the string stability of a set of networked cars.
1. World Health Organization (WHO). Global Status Report on Road Safety 2018. December 2018.
Objectives
1. Analyze the current methods of string stability among homogenous vehicles.
Two of the main approaches are:
a. H∞ Control for String Stability (The focus of this report)
b. Distributed implementation of receding horizon control is formulated to address vehicle
platooning.( Part of literature review)
2. Based on preliminary results design and implement a new method (or new
system formulation) for heterogeneous vehicles.
System Dynamics
In longitudinal vehicle dynamics, it is assumed that the input of the vehicle is the
throttle system (acceleration) while the output of the vehicle is the position.
System Dynamics Transfer Function
It is to be noted that all vehicles(m) in a platoon are
assumed to be identical in this study. Therefore, the
system can be considered homogeneous.
String Stability
● The string stability of plantoons of vehicles is evaluated with the propagation
of the system responses along a cascade of systems. The string stability is
proved based-on asymptotic stability of interconnected vehicles by focusing
on perturbations on initial conditions.
● The string stability is quantified by the magnitude of the string stability transfer
function as follows where ∆
i
is a signal of interest. A condition on the maximal
amplification of perturbation along string in a platoon can be considered as
the requirement of the string stability.
Two Types of Traditional Control Structures
1. ACC (Adaptive Cruise Control)
2. CACC (Cooperative Adaptive Cruise Control)
ACC (Adaptive Cruise Control)
● employs forward-looking sensors, i.e., radar, for longitudinal dynamics control
of the vehicle by using throttle and brake actuations
● The main objective of its structure is to follow the preceding vehicle at a
desired relative distance (p)
● a feedback controller is used to control the spacing error (e
i
) between the
actual and desired distances using a velocity-dependent spacing policy (H
i
)
○ When the spacing error is positive, acceleration is required.
○ When the spacing error is negative, braking is required.
Sensitivity Transfer Function for ACC
● The controller parameters are designed to match the plant dynamics. Therefore, it
is critical that the parameters are chosen in such a way that the closed loop
system is not sensitive to variations in plant dynamics.
● The closed-loop sensitivity transfer function (S
i
) and the closed-loop
complementary sensitivity transfer function (T
i
) are written as follows:
CACC (Cooperative Adaptive Cruise Control)
● In CACC systems, the acceleration of the preceding vehicle (a
i−1
) is obtained
by wireless communication and used as a feed-forward control action for the
following vehicle i. If there is no communication, the system becomes a
traditional ACC system. In addition to using a velocity-dependent spacing
policy (H
i
),
○ since the acceleration is obtained through wireless communication, CACC includes a
communication delay D
i
(s) and a feedforward filter 1/H
i
(s).
Sensitivity Transfer Function for CACC
● Just like the ACC, the CACC aims to ensure that the closed loop system is not
sensitive to variations in plant dynamics.
● The closed-loop sensitivity transfer function (S
i
) and the closed-loop
complementary sensitivity transfer function (T
i
) are written as follows:
Two Types of H∞ Control Structures
1. H∞ with Augmented ACC Plant
2. H∞ with Augmented CACC Plant
H∞ Controller
H
∞
(i.e. "H-infinity") methods are used to synthesize controllers to achieve stabilization with guaranteed performance. To use H
∞
methods, a
control designer expresses the control problem as a mathematical optimization problem and then finds the controller K that solves this
optimization.
● The plant P has two inputs, the exogenous input r, that includes reference signal and disturbances, and the manipulated variables u.
● There are two outputs, y
1
which includes the error signal that we want to minimize, and the measured variables y
2
, that we use to control the
system.
● y
2
is used in K to calculate the manipulated variables u.
● Γ is the best achieved value for the closed-loop H∞ norm
● W
s
, W
u
and W
t
are respectively weighting
functions that penalize respectively the error signal, control signal and output signal,
CACCACC
Multi-Objective H∞ Controller String Stability
● The augmented plant is formulated considering the H∞
control optimization problem.
● An additional constraint condition is added on the
complementary sensitivity function (T) to guarantee the
string stability.
ACC
CACC
Simulation Parameters
● The time-constant = 0.1 seconds.
● The time- delay of the system = 0.2 seconds.
● The communication delay between vehicles in a platoon = 0.15 seconds.
● Choosing the weighting functions were inspired by intuition of the frequency
response domain. We choose a weighting function which will map it into the
unit circle in Z-domain. This guarantees the stability in each control iteration.
● For the spacing policy H
i
we used different real world time policies h
i
ranging
from 0.5 to 4.0 seconds with 0.5 increments.
Result
Result
Result
Result
Discussion
● As shown in the frequency domain response, the complementary sensitivity
function T ≤ 0 dB satisfying the constrained optimization problem.
● Thus, it is concluded that the string stability of the system, both for ACC and
CACC is achieved for different real world spacing policies ranging from 0.5 to
4.0 seconds.
Challenges
● Heterogeneous case: What If we have different types of vehicles which will
result a totally different dynamics and variable transfer functions such as
truck, motorbike…
● We did not consider fuel consumption which is a vital objective/cost function
to optimize in the real world.
● Other control structure designs (different than H-infinity)
References
1. G. J. L. Naus, R. P. A. Vugts, J. Ploeg, M. J. G. van de Molengraft and M. Steinbuch, "String-Stable CACC Design and Experimental
Validation: A Frequency-Domain Approach," in IEEE Transactions on Vehicular Technology, vol. 59, no. 9, pp. 4268-4279, Nov. 2010, doi:
10.1109/TVT.2010.2076320.
2. J. Ploeg, B. T. M. Scheepers, E. van Nunen, N. van de Wouw and H. Nijmeijer, "Design and experimental evaluation of cooperative adaptive
cruise control," 2011 14th International IEEE Conference on Intelligent Transportation Systems (ITSC), Washington, DC, 2011, pp. 260-265,
doi: 10.1109/ITSC.2011.6082981
3. Shuo Feng, Yi Zhang, Shengbo Eben Li, Zhong Cao, Henry X. Liu, Li Li, String stability for vehicular platoon control: Definitions and analysis
methods, Annual Reviews in Control, Volume 47, 2019, Pages 81-97, ISSN 1367-5788, https://doi.org/10.1016/j.arcontrol.2019.03.001.
4. H. Xing, J. Ploeg, and H. Nijmeijer, “Pade; approximation of delays in cooperative acc based on string stability requirements,” IEEE
Transactions on Intelligent Vehicles, vol. PP, no. 99, pp. 1–1, 2017.
5. E. Kayacan, "Multiobjective H infinity Control for String Stability of Cooperative Adaptive Cruise Control Systems," in IEEE Transactions on
Intelligent Vehicles, vol. 2, no. 1, pp. 52-61, March 2017, doi: 10.1109/TIV.2017.2708607.
